Degree matrix

Definition

The degree matrix of a graph is a diagonal matrix where the rows and columns are indexed by the set of vertices (in the same order), and each diagonal entry gives the degree of the corresponding vertex.

In words:

Suppose $G$ is a finite undirected graph. Let $n$ be the size of the vertex set $V(G)$. Fix a bijective correspondence $v:\{1,2,\dots ,n\}\to V(G)$. The degree matrix of $G$ is a $n\times n$ diagonal matrix such that:

• For $i\neq j$ with $1\leq i,j\leq n$, the $(i,j)^{th}$ entry of the matrix is 0.
• For $1\leq i\leq n$ the $(i,i)^{th}$ (diagonal) entry of the matrix is the degree of the vertex $v(i)$.

Note that this matrix is well defined only after we fix the bijection $v$. Changing the bijection has the effect of conjugating the degree matrix by a permutation matrix obtained by composing one bijection with the inverse of the other. In other words, we re-label the rows and re-label the columns in the same manner. In particular, the characteristic polynomial as well as the multiset of eigenvalues for the degree matrix does not depend on the ordering of vertices chosen.

Related notions

• The degree sequence of a graph is informationally equivalent to the degree matrix. The former stores the degrees as a non-increasing sequence of degrees, while the latter uses a matrix structure for storing the information.